Algorithm 1 GaussianSampler

Input:  Lattice $\Lambda$ basis B, standard deviation $\sigma$, center $c\in {\mathbb{Z}}^N$ Output:  Vector v sampled in $D_{\Lambda ,\sigma ,c}$ 1:$v_n\leftarrow 0$ 2:$c_n\leftarrow c$ 3:for  $i=n,n-1,\cdots ,1$  do 4:    $c_i^{{}^{\prime }}\leftarrow \langle c_i,\widetilde{b_i}\rangle /{\Vert \widetilde{b_i}\Vert }^2$ 5:    ${\sigma }_i^{{}^{\prime }}\leftarrow \Vert \widetilde{b_i}\Vert$ 6:    $z_i\leftarrow SampleZ(c_i^{{}^{\prime }},{\sigma }_i^{{}^{\prime }})$ 7:    $c_{i-1}\leftarrow c_i-z_i{b}_i$ 8:    $v_{i-1}\leftarrow v_i-z_i{b}_i$ 9:end for 10:return  $v_0$